Question: Simplify; express your answer in exponential form. Assume $r\neq 0, n\neq 0$. $\dfrac{{(r^{3})^{5}}}{{(rn^{5})^{-2}}}$
Solution: To start, try working on the numerator and the denominator independently. In the numerator, we have ${r^{3}}$ to the exponent ${5}$ . Now ${3 \times 5 = 15}$ , so ${(r^{3})^{5} = r^{15}}$ In the denominator, we can use the distributive property of exponents. ${(rn^{5})^{-2} = (r)^{-2}(n^{5})^{-2}}$ Simplify using the same method from the numerator and put the entire equation together. $\dfrac{{(r^{3})^{5}}}{{(rn^{5})^{-2}}} = \dfrac{{r^{15}}}{{r^{-2}n^{-10}}}$ Break up the equation by variable and simplify. $\dfrac{{r^{15}}}{{r^{-2}n^{-10}}} = \dfrac{{r^{15}}}{{r^{-2}}} \cdot \dfrac{{1}}{{n^{-10}}} = r^{{15} - {(-2)}} \cdot n^{- {(-10)}} = r^{17}n^{10}$.